What are the standard models used to forecast asset price movements? For example, if I were to trade an option, what model would I use in conjunction with option pricing models to forecast the stock price? Should I use GARCH or one of its variants?

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You can forecast stock prices thru time-series models, cross-sectional, or panel models. There is considerable variation within these categories.

In time-series models you would use an auto-regressive model such as an AR(1) where the independent variable is the dependent variable lagged by one period. Naturally, an AR(2) would consist of 2 lags and so on. AR(1) models are best when the stock price exhibits mean reversion and does not have a unit root. You would have to perform an analysis to determine the appropriate # of lags (partial auto-correlation coefficient, checks for seasaonlity, heteroskedasticity, etc.) and so forth.

Other time-series models include the Moving Average MA(1), or combination models such as ARIMA. These models have different properties from AR(1) models. For example, the auto-correlation function of the residuals for a MA(1) collapses to zero after one lag.

GARCH and ARCH models are also worth exploring, particularly for volatility modeling, however this specification is not used for modeling returns because returns do not have clustering and memory in the same way that volatility does.

I would recommend Ruey Tsay's text to learn more about time-series models.

Cross-sectional models use multiple regression to predict returns based on a security's characteristics or exposures to factors. In this case you may elect to explain security returns in terms of a factor model.

Other models might consist of panel models (i.e. time series and cross-sectional) such as random effects or mixed effects.

In any modeling choice, however, you are making a claim as to the structure of the data generating process. I would argue that you begin with empirical study, develop a theory that describes returns, and then formulate your ideas on the best approach to model the series. Some popular theories are covered in John Cochrane's asset pricing text.